Experimentally revealing anomalously large dipoles in the dielectric of a quantum circuit

Quantum two-level systems (TLSs) intrinsic to glasses induce decoherence in many modern quantum devices, such as superconducting qubits. Although the low-temperature physics of these TLSs is usually well-explained by a phenomenological standard tunneling model of independent TLSs, the nature of these TLSs, as well as their behavior out of equilibrium and at high energies above 1 K, remain inconclusive. Here we measure the non-equilibrium dielectric loss of TLSs in amorphous silicon using a superconducting resonator, where energies of TLSs are varied in time using a swept electric field. Our results show the existence of two distinct ensembles of TLSs, interacting weakly and strongly with phonons, where the latter also possesses anomalously large electric dipole moment. These results may shed new light on the low temperature characteristics of amorphous solids, and hold implications to experiments and applications in quantum devices using time-varying electric fields.


I. ADDITIONAL DETAILS OF THE EXPERIMENTAL SETUP
The measurements are performed in a dilution refrigerator at a temperature of less than 20 mK. Supplementary Figure 1 shows the schematic of the detailed wirings of the measurement setup. A vector network analyzer (VNA) is used to carry out the S 21 measurements. The input line is heavily filtered with multiple attenuators at different stages.
An additional 12 GHz K&L low-pass filter is added. The output signal is routed through two low-noise circulators (4 − 12 GHz) which are thermally anchored at mixing chamber plate. The output signal is amplified by a high electron mobility transistor (HEMT) at 4 K. For the bias line, a −10 dB attenuator is added at room temperature. A 12 GHz K&L low-pass filter and a RC filter are placed at mixing chamber stage, which gives a cut-off frequency of 10 MHz. A time-varying bias voltage as a triangular waveform is applied. The maximum bias is E b = 0.44 V/µ m, and the fastest applied bias frequency is f b = 4.5 MHz. In previous related measurements, the time-dependent loss, particularly when the bias rate changes sign, was studied [1]. It is known that the change to the measured average loss is negligible.
The base temperature of the dilution refrigerator remains below 20 mK throughout all measurements, such that no significant heating from bias leakage current is observed.
1. Schematic of the experimental wiring for measuring the resonator.

II. INPUT MICROWAVE POWER CALIBRATION
To determine the input power on chip due to attenuation in the input line, throughput S 21 's (bypassing the mounted device) at various source powers have been measured at room temperature. This is particularly important to accurately determine the average photon number in the resonator. The attenuation depends on the signal frequency and the source power from the VNA. Four relatively high source power (10 dBm, 0 dBm, −10 dBm and −20 dBm) are used to obtain sufficient signal to noise ratio. As shown in Supplementary Fig. 2a, linear fits are conducted on all four semilog S 21 plots. The frequency dependence is obtained by averaging the four slopes at different applied source powers. As shown in Supplementary Fig. 2b, the source power dependence of the input power at zero frequency is determined by plotting the intercepts as a function of the source powers. The obtained slope is one, which suggests any nonlinearities in the system associated with the power is negligible. Therefore, the eventual input power at mixing chamber can be rotation from a small impedance mismatch. We plot the data with adjusted amplitude and electric delays on the right.
The quality factor extraction is following the fitting function[2] ( The Amp and θ are the adjusted parameters for the raw data due to the attenuations, amplifications, and electric delays in both input and output line, and Q is the total quality factor. Because of the impedance mismatch at two ports,Q e is the external quality factor and also a complex number, and ϕ is the angle ofQ −1 e . Fig. 3a shows an exemplary S 21 data (magnitude vs frequency) measured at P ac = −29 dBm. The corresponding IQ plot of the data are shown in Fig. 3b: The lower left curve (blue dots) and the right curve (blue dots) are data before and after the adjusted Amp and θ. From the fitting, we extract Q e = 1/Re{Q −1 e } ≈ 7300, Q i ≈ 9000, and ϕ = 5.7 • when the input photon number n = 1422.